Question: Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{-3}t^{-2})^{-4}}}{{(a^{2}t^{2})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{-3}t^{-2})^{-4} = (a^{-3})^{-4}(t^{-2})^{-4}}$ On the left, we have ${a^{-3}}$ to the exponent ${-4}$ . Now ${-3 \times -4 = 12}$ , so ${(a^{-3})^{-4} = a^{12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{-3}t^{-2})^{-4}}}{{(a^{2}t^{2})^{-3}}} = \dfrac{{a^{12}t^{8}}}{{a^{-6}t^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{12}t^{8}}}{{a^{-6}t^{-6}}} = \dfrac{{a^{12}}}{{a^{-6}}} \cdot \dfrac{{t^{8}}}{{t^{-6}}} = a^{{12} - {(-6)}} \cdot t^{{8} - {(-6)}} = a^{18}t^{14}$